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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.11205 |
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| _version_ | 1866911153537417216 |
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| author | De Filippis, Cristiana |
| author_facet | De Filippis, Cristiana |
| contents | Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sketches of Nonuniformly Elliptic Schauder Theory De Filippis, Cristiana Analysis of PDEs Schauder theory is a basic tool in the study of elliptic and parabolic PDEs, asserting that solutions inherit the regularity of the coefficients. It plays a central role in establishing higher regularity for solutions to a broad class of elliptic problems exhibiting ellipticity, including those involving free boundaries. In the linear setting, Schauder theory dates back to the 1920-30s and is now considered classical. Nonlinear extensions were developed in the 1980s. All these classical results are restricted to uniformly elliptic operators and heavily rely on perturbative techniques - freezing the coefficients and comparing the solution to that of a constant-coefficient problem. However, such methods fail in the nonuniformly elliptic setting, where homogeneous a priori estimates break down and standard iteration arguments no longer apply. Here we give a brief survey on recent progresses including the solution to the longstanding problem of proving the validity of Schauder estimates in the nonlinear, nonuniformly elliptic setting. |
| title | Sketches of Nonuniformly Elliptic Schauder Theory |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.11205 |